The Nested Interval Theorem

Theorem.If
is a sequence of nested, closed, bounded, nonempty intervals,
then
is nonempty. If, in addition,
then
consists of a single point.

Proof.
Write
The fact that we have nested intervals means

Thus, the form an
increasing, bounded (above) sequence, while the
form a
decreasing, bounded (below) sequence.

By
a familiar fact
from calculus,
every
monotone, bounded sequence converges.
Thus, the
converge to some number
while the
converge to some number
which satisfy
.
Clearly, both a and b are elements of
,
because both are an elements of the closed interval
for any n. (Why?)
In fact, it's not hard to see that
is precisely the interval [a, b].
Finally, if
then we have a = b; that is,
= {a}.